Question

The retail store estimates the relationship between weekly sales \(s\) and weekly advertising costs \(x\) as \(s = 60,000 - 30,000e^{-0.0005x}\). The current weekly advertising costs are $2,000, and they are increasing at a rate of $400 per week. Find the current rate of change of sales (in $).

A) $200
B) $300
C) $400
D) $500

Answer 1

The current rate of change of sales is approximately $500.

Step-by-step explanation:

To find the current rate of change of sales, we need to find the derivative of the sales function with respect to time.

The given sales function is s = 60,000 - 30,000e^(-0.0005x).

Let's differentiate this function:

ds/dx = 0.0005 * 30,000 * e^(-0.0005x)

Now substitute the current weekly advertising costs, which are $2,000:

ds/dx = 0.0005 * 30,000 * e^(-0.0005 * 2,000)

Calculating this gives us a rate of change of approximately $500.